On the Amplitude Equations Arising at the Onset of the Oscillatory Instability in Pattern Formation

A well-known system of two amplitude equations is considered that describes the weakly nonlinear evolution of many nonequilibrium systems at the onset of the so-called oscillatory instability. Those equations depend on a small parameter, ε, that is a ratio between two distinguished spatial scales. In the limit ε → 0, a simpler asymptotic model is obtained that consists of two complex cubic Ginzburg–Landau equations, coupled only by spatially averaged terms.